240 XIII INTEGRATION
(cf. Problem 1). For the general case, consider the functions sup(w, f<pKn), where
(Kn) is an increasing sequence of compact sets whose union is X.)
20. Let U be a continuous endomorphism of the space Li(X, /it) satisfying the hypotheses
of Section 13.11, Problem 17, so that U extends to an endomorphism of each of the
spaces L£(X, JLC), 1 <>p <; -f oo (he. cit.). For each function fe && , put
R«(/) = - (/+#•/+••' + U"'1 •/),
n
R*(/) = sup |Rk(/)|,
l^k^n
R*(/) = supR*(/).
n
(a) Show that if 1< p < + oo and / e -2=$ , then R*(/) e -2* and
p,.
P— I
(Use Problem 19 above, and Section 13.11, Problem 17(c).)
(b) For each function/ e J§?£, let P -/be the limit in Lj| of the sequence ((Rn(/))~)
(Section 12.15, Problem 12(c)). We have £/P = P(7 = P. Show that the sequence
(Rn(/)) converges almost everywhere to P -/(Dunford-Schwartz ergodic theorem).
(Put
S(/) = lim sup Rn(/),
rt-»OO
= liminfRM(/).
Observe that !(/) g S(/) ^ R*(/), and that S(Rm(/) -P-/) = S(/)-P-/ and
I(Rm(/) -P */) = K/) -P -/for all m £ 1, and use (a) with Rm(/) -P • /in place
of/)
(c) Show that N^P •/) £ NI(/) for all / e && n J2?ff , and hence that P extends to a
contraction on the space LR. Deduce that the sequence (R»(/)) converges almost
everywhere to/, for each function /e jfifj. (Put
L(/) = limsup|Rw(/)-P-/|,
n-*oo
observe that L(/) ^ R*(/-^) + P • \f-g\ for all ^ e Se\ n jgff , and make use of
Section 13.11, Problem 7(d).)
21. The notation is that of Section 13.17, Problem 7. For each a > 0 and each function
g e &}OCtC(y, v), let AM denote the set of points y e Y such that \g(y)\ > a. If
p e [1, + oo [, the endomorphism U is said to be of weak type (p,p) if there exists a
constant C > 0 such that, for each fc-integrable step function / and each a > 0, we
have and that